YMSC seminar on complex algebraic geometry | Specialization of Linear Differential Equations

Abstract:

In this talk, we will discuss recent advancements in the specialization of linear differential equations. Given a linear differential equation with parameters, our focus is on how the algebraic properties of its solutions change as these parameters vary over an affine variety. For example, we inquire about the values of parameters for which linear differential equations can have a basis of algebraic solutions, assuming the original equation lacks such a basis. By extending a result of Hrushovski, we demonstrate that the set of such values is indeed ``small" in a meaningful sense. As an application, we establish Matzat’s conjecture in its entirety: The absolute differential Galois group of a one-variable function field, equipped with a non-trivial derivation, is the free proalgebraic group. The talk is based on joint work with Michael Wibmer from the University of Leeds, UK.